The existence and uniqueness results of fully coupled forward-backward stochastic differential equations with stopping time (unbounded) is obtained. One kind of comparison theorem for this kind of equations is also pr...The existence and uniqueness results of fully coupled forward-backward stochastic differential equations with stopping time (unbounded) is obtained. One kind of comparison theorem for this kind of equations is also proved.展开更多
This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information avail...This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.展开更多
This paper studies the well-posedness of fully coupled linear forward-backward stochastic differential equations (FBSDEs). The authors introduce two main methods-the method of continuation under monotonicity condition...This paper studies the well-posedness of fully coupled linear forward-backward stochastic differential equations (FBSDEs). The authors introduce two main methods-the method of continuation under monotonicity conditions and the unified approach-to ensure the existence and uniqueness of solutions of fully coupled linear FBSDEs. The authors show that the first method (the method of continuation under monotonicity conditions) can be deduced as a special case of the second method (the unified approach). An example is given to illustrate it in linear FBSDEs case. And then, a linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs can not be dealt with by the the method of continuation under monotonicity conditions and the unified approach directly. As a powerful supplement to the the method of continuation under monotonicity conditions and the unified approach, linear transformation method overall develops the well-posedness theory of fully coupled linear forward-backward stochastic differential equations which have potential applications in optimal control and partial differential equation theory.展开更多
We propose a novel numerical scheme for decoupled forward-backward stochastic differ- ential equations （FBSDEs） in bounded domains, which corresponds to a class of nonlinear parabolic partial differential equations ...We propose a novel numerical scheme for decoupled forward-backward stochastic differ- ential equations （FBSDEs） in bounded domains, which corresponds to a class of nonlinear parabolic partial differential equations with Dirichlet boundary conditions. The key idea is to exploit the regularity of the solution （Yt,Zt） with respect to Xt to avoid direct ap- proximation of the involved random exit time. Especially, in the one-dimensional case, we prove that the probability of Xt exiting the domain within At is on the order of O（（△t）ε exp（--1/（△t）2ε））, if the distance between the start point X0 and the boundary is 1 g at least on the order of O（△t）^1/2-ε ） for any fixed c 〉 0. Hence, in spatial discretization, we set the mesh size △x - （9（（At）^1/2-ε ）, so that all the interior grid points are sufficiently far from the boundary, which makes the error caused by the exit time decay sub-exponentially with respect to △t. The accuracy of the approximate solution near the boundary can be guaranteed by means of high-order piecewise polynomial interpolation. Our method is developed using the implicit Euler scheme and cubic polynomial interpolation, which leads to an overall first-order convergence rate with respect to △t.展开更多
We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones ...We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18： 143-177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.展开更多
A necessary maximum principle is given for nonzero-sum stochastic Oltterential games with random jumps. The result is applied to solve the H2/H∞ control problem of stochastic systems with random jumps. A necessary an...A necessary maximum principle is given for nonzero-sum stochastic Oltterential games with random jumps. The result is applied to solve the H2/H∞ control problem of stochastic systems with random jumps. A necessary and sufficient condition for the existence of a unique solution to the H2/H∞ control problem is derived. The resulting solution is given by the solution of an uncontrolled forward backward stochastic differential equation with random jumps.展开更多
基金This work was supported by the National Natural Science Foundation of China (10001022 and 10371067)the Excellent Young Teachers Program and the Doctoral program Foundation of MOE and Shandong Province,P.R.C.
文摘The existence and uniqueness results of fully coupled forward-backward stochastic differential equations with stopping time (unbounded) is obtained. One kind of comparison theorem for this kind of equations is also proved.
文摘This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.
基金supported by the National Natural Science Foundation of China under Grant No.61573217the National High-Level Personnel of Special Support Programthe Chang Jiang Scholar Program of Chinese Education Ministry
文摘This paper studies the well-posedness of fully coupled linear forward-backward stochastic differential equations (FBSDEs). The authors introduce two main methods-the method of continuation under monotonicity conditions and the unified approach-to ensure the existence and uniqueness of solutions of fully coupled linear FBSDEs. The authors show that the first method (the method of continuation under monotonicity conditions) can be deduced as a special case of the second method (the unified approach). An example is given to illustrate it in linear FBSDEs case. And then, a linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs can not be dealt with by the the method of continuation under monotonicity conditions and the unified approach directly. As a powerful supplement to the the method of continuation under monotonicity conditions and the unified approach, linear transformation method overall develops the well-posedness theory of fully coupled linear forward-backward stochastic differential equations which have potential applications in optimal control and partial differential equation theory.
基金The authors would like to thank the referees for their valuable comments, which have improved the quality of the paper. This work is partially supported by the National Natural Science Foundations of China under grant numbers 91130003, 11171189 and 11571206 and by Natural Science Foundation of Shandong Province under grant number ZR2011AZ002+2 种基金 the U.S. Defense Advanced Research Projects Agency, Defense Sciences Office under contract HR0011619523 the U.S. Department of Energy, Office of Science, Office of Advanced ScientificComputing Research, Applied Mathematics program under contracts ERKJ259, ERKJ320 the U.S. National Science Foundation, Computational Mathematics program under award 1620027.
文摘We propose a novel numerical scheme for decoupled forward-backward stochastic differ- ential equations （FBSDEs） in bounded domains, which corresponds to a class of nonlinear parabolic partial differential equations with Dirichlet boundary conditions. The key idea is to exploit the regularity of the solution （Yt,Zt） with respect to Xt to avoid direct ap- proximation of the involved random exit time. Especially, in the one-dimensional case, we prove that the probability of Xt exiting the domain within At is on the order of O（（△t）ε exp（--1/（△t）2ε））, if the distance between the start point X0 and the boundary is 1 g at least on the order of O（△t）^1/2-ε ） for any fixed c 〉 0. Hence, in spatial discretization, we set the mesh size △x - （9（（At）^1/2-ε ）, so that all the interior grid points are sufficiently far from the boundary, which makes the error caused by the exit time decay sub-exponentially with respect to △t. The accuracy of the approximate solution near the boundary can be guaranteed by means of high-order piecewise polynomial interpolation. Our method is developed using the implicit Euler scheme and cubic polynomial interpolation, which leads to an overall first-order convergence rate with respect to △t.
基金Acknowledgements The authors would like to thank the referees for the valuable comments, which improved the paper a lot. This work was partially supported by the National Natural Science Foundations of China （Grant Nos. 91130003, 11171189） and the Natural Science Foundation of Shandong Province （No. ZR2011AZ002）.
文摘We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18： 143-177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.
基金supported by the Doctoral foundation of University of Jinan(XBS1213)the National Natural Science Foundation of China(11101242)
文摘A necessary maximum principle is given for nonzero-sum stochastic Oltterential games with random jumps. The result is applied to solve the H2/H∞ control problem of stochastic systems with random jumps. A necessary and sufficient condition for the existence of a unique solution to the H2/H∞ control problem is derived. The resulting solution is given by the solution of an uncontrolled forward backward stochastic differential equation with random jumps.
